What is Paris Law for Fatigue?

The Paris Law, also known as the Paris-Erdogan equation, is a fundamental model used in fracture mechanics to predict the rate at which a fatigue crack grows in a material under cyclic loading. It relates the crack growth rate (da/dN) to the range of the stress intensity factor (ΔK). This tool allows engineers and students to estimate how much a crack will extend over a specific number of load cycles, which is critical for damage-tolerant design and structural health monitoring.

The Mathematics Behind the Calculation

The primary formula is: da/dN = C(ΔK)^m. Here, ΔK is determined by the stress range (Δσ), the crack length (a), and a geometry factor (Y) using the relation: ΔK = Y · Δσ · √(πa). Because the crack length 'a' changes as the cycles progress, simple multiplication isn't enough; we must integrate the equation. This calculator uses the analytical integration for constant amplitude loading, providing an estimate of the crack size after a set number of cycles.

How to Use the Paris Law Calculator

To use this calculator effectively, you need five main inputs: the current crack length, the cyclic stress applied to the part, the geometry factor (often 1.12 for surface cracks), and the material constants C and m. Material constants are typically found in engineering handbooks (e.g., for steel, m is often around 3.0). Input your values and the number of cycles you wish to simulate, and the tool will output the predicted final crack length in millimeters.

Frequently Asked Questions

What are the units for C? The units for C depend on the units used for stress and length. This calculator assumes SI units (MPa and mm). If your C value is in different units, you must convert them before inputting.

When is Paris Law valid? Paris Law is applicable in 'Region II' of the fatigue crack growth curve. It does not account for the threshold region (where cracks don't grow) or the final fracture region (where growth is unstable).